$Project Greater Ed Logo$

The Multiplication Rule for Radicals and Factor Trees

Precalculus Algebra Simplifying Radical Expressions The Multiplication Rule for Radicals and Factor Trees

Mini Lecture Video

The Multiplication Rule for Radicals

Theorem

Let $a$ and $b$ be real numbers or variable expressions. Let $n$ be any natural number. Then
$$\sqrt[n]{a\cdot b}=\sqrt[n]{a}\cdot \sqrt[n]{b}$$

In a manner of speaking, radicals can be “distributed” over products (i.e. stuff being multiplied). Similarly, reading the equation in reverse, the product of radicals/roots (when the roots are the same on the outside; i.e. have the same “power”) can be turned into one single radical with the inside stuff being multiplied within that radical.

Using our rule, we can write $\sqrt{4x}$ as

$$\sqrt{4x}=\sqrt{4}\sqrt{x}=2\sqrt{x}$$

and the latter is completely simplified.

Simplifying Radicals by Prime Factorization (Using Factor Trees)

We can use this ability to “distribute radicals,” along with the cancellation rule discussed in the previous lesson to help us reduce radicals of large composite (i.e. non-prime) numbers. The method that follows will help us reduce a radical like $\sqrt{450}$ to

$$\sqrt{450}=15\sqrt{2}$$

The Method

To illustrate, lets consider the following example: $\sqrt{48}$.

Fully factor the number on the inside of the radical into prime factors, grouping them into powers corresponding with the root being taken where possible.
- Factor trees, as demonstrated in the video help with this
- In our example: $\sqrt{48}=\sqrt{2^2\cdot 2^2\cdot 3}$
Distribute the radical to each factor power you see inside the radical, and cancel the radical with the root where possible.
- In our example: $\sqrt{2^2\cdot 2^2\cdot 3}=\sqrt{2^2}\cdot\sqrt{2^2}\cdot \sqrt{3}=2\cdot 2 \cdot \sqrt{3}$
Combine/multiply stuff out in front of any remaining radicals, and then you’re done.
- In our example: $2\cdot 2 \cdot \sqrt{3}=4\sqrt{3}$

The above method works as well for powers of variables inside radicals. Just be careful to use absolute values when cancelling even radicals with even powers!

NOTE: You can certainly skip the “grouping the factors into powers corresponding to the root being taken” step (or at least simplify it a bit) by grouping the prime factors into powers of numbers that are evenly divisible by the power of the root, then dividing the power by the power of the root.

For example: $\sqrt{48}=\sqrt{2^4\cdot 3}=2^{4/2} \sqrt{3}=4\sqrt{3}$

Try the Following!

Hint: Factor $52$ completely, then group factors into powers of 2 where possible.

Solution: $2\sqrt{13}$

Hint: Factor the $81$ completely, then group factors into powers of 3 where possible.

Solution: $3\sqrt[3]{2}$

Solution: $3\sqrt[3]{3}$

Hint: Factor the $256$ completely, then group factors into powers of 4, 8, or 16 (the latter two if you want to skip a step).

Solution: $4$

Solution: $\sqrt[3]{12}$ it can’t be reduced any more. The factors of $12$ don’t have high enough powers to cancel the cube root.

Solution: $-2\sqrt[3]{45}$

Previous Topic

Back to Lesson

Next Topic